1. **Problem statement:** The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to the base is 3 : 2. Using Heron's formula, find the area of the triangle. 2. **Step 1: Define variables and express sides in terms of a variable.** Let the equal sides be $3x$ each and the base be $2x$. 3. **Step 2: Use the perimeter to find $x$.** The perimeter $P = 3x + 3x + 2x = 8x$. Given $P = 32$, so: $$8x = 32$$ $$x = \frac{32}{8} = 4$$ 4. **Step 3: Find the lengths of the sides.** Equal sides: $3x = 3 \times 4 = 12$ cm each. Base: $2x = 2 \times 4 = 8$ cm. 5. **Step 4: Calculate the semi-perimeter $s$.** $$s = \frac{12 + 12 + 8}{2} = \frac{32}{2} = 16$$ 6. **Step 5: Apply Heron's formula for area $A$.** Heron's formula: $$A = \sqrt{s(s - a)(s - b)(s - c)}$$ where $a=12$, $b=12$, $c=8$. 7. **Step 6: Substitute values and simplify.** $$A = \sqrt{16(16 - 12)(16 - 12)(16 - 8)} = \sqrt{16 \times 4 \times 4 \times 8}$$ 8. **Step 7: Calculate the product inside the square root.** $$16 \times 4 = 64$$ $$64 \times 4 = 256$$ $$256 \times 8 = 2048$$ 9. **Step 8: Find the square root.** $$A = \sqrt{2048} = \sqrt{1024 \times 2} = 32 \sqrt{2}$$ 10. **Final answer:** The area of the isosceles triangle is $32 \sqrt{2}$ cm$^2$.